Closed Extension Topology is not Hausdorff

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_p = \left({S^*_p, \tau^*_p}\right)$ be the closed extension space of $T$.


Then $T^*_p$ is not a $T_2$ (Hausdorf) space.


Proof

Aiming for a contradiction, suppose $T^*_p$ is not a $T_2$ (Hausdorf) space.

From $T_2$ Space is $T_1$ Space, $T^*_p$ is a $T_1$ (Fréchet) space.

But this contradicts Closed Extension Topology is not $T_1$

Hence by Proof by Contradiction $T^*_p$ can not be a $T_2$ (Hausdorf) space.

$\blacksquare$


Sources